Early Fault-Tolerant Quantum Algorithms in Practice: Application to Ground-State Energy Estimation

TL;DR

This paper explores early fault-tolerant quantum algorithms for ground-state energy estimation, proposing a signal processing approach to improve efficiency and practicality in resource-constrained quantum computing scenarios.

Contribution

It introduces a novel method for estimating spectral measures without requiring exact ground states, and provides quantitative resource estimates demonstrating efficiency gains.

Findings

Quantum algorithms closely match DMRG energies at larger bond dimensions.

Significant reduction in samples needed compared to theoretical estimates.

CDF-based algorithms are practical alternatives to quantum phase estimation.

Abstract

We investigate the feasibility of early fault-tolerant quantum algorithms focusing on ground-state energy estimation problems. In particular, we examine the computation of the cumulative distribution function (CDF) of the spectral measure of a Hamiltonian and the identification of its discontinuities. Scaling these methods to larger system sizes reveals three key challenges: the smoothness of the CDF for large supports, the lack of tight lower bounds on the overlap with the true ground state, and the difficulty of preparing high-quality initial states. To address these challenges, we propose a signal processing approach to find these estimates automatically, in the regime where the quality of the initial state is unknown. Rather than aiming for exact ground-state energy, we advocate for improving classical estimates by targeting the low-energy support of the initial state. Additionally, we provide quantitative resource estimates, demonstrating a constant-factor improvement in the number of samples required to detect a specified change in CDF. Our numerical experiments, conducted on a 26-qubit fully connected Heisenberg model, leverage a truncated density-matrix renormalization group (DMRG) initial state with a low bond dimension. The results show that the predictions from the quantum algorithm align closely with the DMRG-converged energies at larger bond dimensions while requiring several orders of magnitude fewer samples than theoretical estimates suggest. These findings underscore that CDF-based quantum algorithms are a practical and resource-efficient alternative to quantum phase estimation, particularly in resource-constrained scenarios.